Biochemical dynamics are often determined by series of single molecule events such as gene expression and reactions involving protein concentrations at nanomolar concentrations. Molecular fluctuations, consequently, may be of biological significance. For example, heterogeneity in clonal populations is believed to arise from molecular fluctuations in gene expression. A realistic description, therefore, requires a probabilistic description of the biochemical dynamics as deterministic descriptions cannot capture the inherent molecular fluctuations. The Gillespie algorithm [D. T. Gillespie, J. Phys. Chem. 81, 2350 (1977)] is a stochastic procedure for simulating chemical systems at low concentrations. A limitation of stochastic kinetic models is that they require detailed information about the chemical kinetics often unavailable in biological systems. Furthermore, the Gillespie algorithm is computationally intensive when there are many molecules and reaction events. In this article, we explore one approximation technique, well known in deterministic kinetics, for simplifying the stochastic model: the quasi-steady-state assumption (QSSA). We illustrate how the QSSA can be applied to the Gillespie algorithm. Using the QSSA, we derive stochastic Michaelis–Menten rate expressions for simple enzymatic reactions and illustrate how the QSSA is applied when modeling and simulating a simple genetic circuit.

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Stochastic chemical kinetics and the quasi-steady-state assumption: Application to the Gillespie algorithm

http://keszei.chem.elte.hu/rkinetika/EnzimIrodalom/stochmodelling_invivoreactions.pdf

Review: Stochastic approaches for modelling in vivo reactions

Single-molecule enzymology: stochastic Michaelis-Menten kinetics.

The application of the quasi-steady-state approximation (QSSA) in biochemical kinetics allows the reduction of a complex biochemical system with an initial fast transient into a simpler system. The simplified system yields insights into the behavior of the biochemical reaction, and analytical approximations can be obtained to determine its kinetic parameters. However, this process can lead to inaccuracies due to the inappropriate application of the QSSA. Here we present a number of approximate solutions and determine in which regions of the initial enzyme and substrate concentration parameter space they are valid. In particular, this illustrates that experimentalists must be careful to use the correct approximation appropriate to the initial conditions within the parameter space.

/pdf/a-century-of-enzyme-kinetics-reliability-of-the-k-m-and-v-v-52o5fey54z.pdf

I.G. Darvey

Papers

The application of the theory of Markov processes to the reversible one substrate-one intermediate-one product enzymic mechanism.

Integrated steady-state rate equations for enzyme-catalyzed reactions.

A new method for the derivation of rate equations in enzyme kinetics using the maximum rate of product formation

The use of power series solutions in the kinetics of enzyme reactions with one intermediate

The determination of the number of intermediates in enzyme catalysed reactions